Theorem cotr3 15138

Description: Two ways of saying a relation is transitive. (Contributed by RP, 22-Mar-2020.)

Hypotheses

Ref	Expression
cotr3.a	⊢ 𝐴 = dom 𝑅
cotr3.b	⊢ 𝐵 = (𝐴 ∩ 𝐶)
cotr3.c	⊢ 𝐶 = ran 𝑅

Assertion

Ref	Expression
cotr3	⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝑧,𝐶

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥, 𝑦)

Proof of Theorem cotr3

Step	Hyp	Ref	Expression
1		cotr3.a	. . 3 ⊢ 𝐴 = dom 𝑅
2	1	eqimss2i 3970	. 2 ⊢ dom 𝑅 ⊆ 𝐴
3		cotr3.b	. . . 4 ⊢ 𝐵 = (𝐴 ∩ 𝐶)
4		cotr3.c	. . . . 5 ⊢ 𝐶 = ran 𝑅
5	1, 4	ineq12i 4114	. . . 4 ⊢ (𝐴 ∩ 𝐶) = (dom 𝑅 ∩ ran 𝑅)
6	3, 5	eqtri 2783	. . 3 ⊢ 𝐵 = (dom 𝑅 ∩ ran 𝑅)
7	6	eqimss2i 3970	. 2 ⊢ (dom 𝑅 ∩ ran 𝑅) ⊆ 𝐵
8	4	eqimss2i 3970	. 2 ⊢ ran 𝑅 ⊆ 𝐶
9	2, 7, 8	cotr2 15137	1 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐶 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∀wral 3076   ⊆ wss 3896   ∩ cin 4088   class class class wbr 5102  dom cdm 5650  ran crn 5651   ∘ ccom 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661

This theorem is used by: (None)